REGULARITY FOR THE STATIONARY NAVIER-STOKES EQUATIONS OVER BUMPY BOUNDARIES AND A LOCAL WALL LAW

REGULARITY FOR THE STATIONARY NAVIER-STOKES EQUATIONS OVER BUMPY BOUNDARIES AND A LOCAL WALL LAW

MITSUO HIGAKI^∗AND CHRISTOPHE PRANGE

Abstract. We investigate regularity estimates for the stationary Navier-Stokes equations above a highly oscillating Lipschitz boundary with the no-slip boundary condition. Our main result is an improved Lipschitz regularity estimate at scales larger than the boundary layer thickness. We also obtain an improved C

estimate and identify the building blocks of the regularity theory, dubbed ‘Navier polynomials’. In the case when some structure is assumed on the oscillations of the boundary, for instance periodicity, these estimates can be seen as local error estimates. Although we handle the regularity of the nonlinear stationary Navier-Stokes equations, our results do not require any smallness assumption on the solutions.

Keywords Navier-Stokes equations, homogenization, boundary layers, compactness me- thods, uniform Lipschitz estimates, improved regularity, large-scale regularity, wall laws, effective boundary conditions

Mathematics Subject Classification (2010) 35B27 · 35B65 · 35Q30 · 76D03 · 76D05 · 76D10 · 76M50

1. INTRODUCTION

This paper is concerned with the local regularity of viscous incompressible fluid flows above rough bumpy boundariesx3> εγ(x⁰/ε)withγLipschitz and the no-slip boundary condition. Al- though bumpy boundaries have a complicated geometry and low regularity, the flow may paradox- ically be better behaved than for smooth or flat boundaries. It is well documented in the physical [29,45] and the mathematical [28,40,20,25] literature that roughness favors slip of the fluid on the boundary in certain regimes. In the striking paper [15] it is even showed experimentally that roughness may delay the transition to turbulence. This also supports the idea that the vanishing vis- cosity limit from Navier-Stokes to Euler may be less singular above highly oscillating boundaries than above flat ones [26,19,43].

Our goal is to investigate these effects, such as the enhanced slip, or the delay of the transition to turbulence, from the point of view of the regularity theory. Due in particular to vorticity creation at the boundary, the boundary regularity of fluid flows with the no-slip boundary conditions is delicate.

In the nonstationary case, it is for instance not known whether there is an analogue of Constantin and Fefferman’s [13] celebrated geometric regularity criteria for supercritical blow-up scenarios. For perfect slip or Navier-slip boundary conditions on the contrary, the situation is brighter. In particular an extension of the criteria of [13] is known in this case; see the work [11] by Beir˜ao da Veiga and Berselli and [38] by Li. We expect that fluids over bumpy boundaries have an intermediate behavior between these two extreme no-slip and (full-)slip situations, especially as far as the mesoscopic regularity properties are concerned.

Our approach grounds on the use of asymptotic analysis to prove regularity estimates. The success of such methods to prove the regularity to certain Partial Differential Equations is spectacular. One

Date: November 28, 2019.

∗Part of this work was done while the first author was a postdoctoral researcher at Universit´e de Bordeaux.

1

of the striking examples is that of homogenization. The basic idea is that the large-scale regularity is determined by the macroscopic properties of the systems, i.e. in the homogenization limit, while the small-scale regularity is determined by the regularity of the data (coefficients, boundary). Two approaches were developed: (a) blow-up and compactness arguments in periodic homogenization in the wake of the pioneering works [8,9], (b) quantitative arguments based on suboptimal local error estimates as developed for periodic homogenization [47,17,44], almost periodic homogenization [7], and stochastic homogenization [23,5].

In this work, we focus on the regularity for stationary problems. We consider the three-dimensional stationary Navier-Stokes equations

(NS^ε)







−∆u^ε+∇p^ε=−u^ε· ∇u^ε inB_1,+^ε (0)

∇ ·u^ε= 0 inB_1,+^ε (0) u^ε= 0 onΓ^ε₁(0),

where the functionsu^ε = u^ε(x) = (u^ε₁(x), u^ε₂(x), u^ε₃(x))^> ∈ R³ andp^ε = p^ε(x) ∈ Rdenote respectively the velocity field and the pressure field of the fluid. We have set forε ∈ (0,1]and r∈(0,1],

B_r,+^ε (0) ={x∈R³|x⁰ ∈(−r, r)², εγ(x⁰

ε)< x₃< εγ(x⁰ ε) +r}, Γ^ε_r(0) ={x∈R³|x⁰ ∈(−r, r)², x3=εγ(x⁰

ε)}. (1)

The boundary functionγ∈W^1,∞(R²)is assumed to satisfyγ(x⁰)∈(−1,0)for allx⁰∈R². Our use of compactness arguments to tackle the regularity for solutions of (NS^ε) is reminiscent of the pioneering work of Avellaneda and Lin [8,9] in homogenization, and of the works by G´erard- Varet [18], Gu and Shen [24], and Kenig and Prange [31,32]. We separate the small-scale regularity, i.e. at scales.ε, from the mescopic- or large-scale regularity, i.e. at scalesε.r≤1. Concerning the small scales, the classical Schauder regularity theory for the Stokes and the Navier-Stokes equa- tions was started by Ladyˇzenskaja [36] using potential theory and by Giaquinta and Modica [22]

using Campanato spaces. These classical estimates require some smoothness of the boundary and typically depend on the modulus of continuity of∇γwhen the boundary is given byx3 =γ(x⁰).

Therefore, these estimates degenerate for highly oscillating boundariesx3 =εγ(x⁰/ε)with suffi- ciently smallε ∈ (0,1). As for the large scales, on the contrary, the regularity is inherited from the limit system whenε→0posed in a domain with a flat boundary. Here no regularity is needed for the original boundary, beyond the boundedness ofγand of its gradient. The mechanism for the regularity at small scales and at large scales is hence completely different. Moreover, it is possible to prove, at the large scales, improved estimates that are known to be false at the small scales. An example of this is our large-scale Lipschitz estimate of Theorem1below that is known to be false over a Lipschitz graph at the small scales even in the case of a linear elliptic operator [33,34,47].

Beyond improved regularity estimates, our objective is to develop local error estimates for the homogenization of viscous incompressible fluids over bumpy boundaries and derive local wall laws.

The wall law catches an averaged effect from theO(ε)-scale on large scale flows of orderO(1) through homogenization. In the wall law, a rough boundary is modeled as a smooth one and an appropriate condition is imposed on it reflecting the roughness of the original boundary. In typical situations, this process gives a Navier-type condition with slip length ofO(ε), the so-called Navier wall law. This effective boundary condition reads for instance in two dimensions

(2) u₁=εα∂₂u₁, u₂= 0 on ∂R²+

with a constantαdepending only on the boundary functionγ. We now briefly review the literature concerned with the derivation of wall laws such as (2) and the proof of error estimates in the global setting. The literature is vast and it is impossible to be exhaustive here. The wall law for simple sta- tionary shear flows is analyzed in the pioneering work J¨ager and Mikeli´c [27] when the boundary is periodic. This result is extended to a random setting by G´erard-Varet [18] and to the almost periodic setting by G´erard-Varet and Masmoudi [20]. Nonstationary cases are studied in Mikeli´c, Ne˘casov´a, and Neuss-Radu [41] under the assumption that the limit flows are space-timeC²functions. The strong regularity condition in [41] implies that a careful analysis is needed when we study Initial

Boundary Value Problems (IBVPs). Indeed, for these cases, no matter how regular the initial data are, there is the loss of regularity of solutions due to the boundary compatibility condition. Higaki [25] considers an IBVP in a bumpy half-space and verifies the Navier wall law forC¹ initial data under natural compatibility conditions. A key ingredient is to make use of theL^∞-regularity theory of the Navier-Stokes equations in the half-spaces; see Abe and Giga [1] for the analyticity of the Stokes semigroup in theL^∞-type spaces. Theorem2 below provides a local counterpart of these global error estimates in the case of the stationary Navier-Stokes equations.

Outline and novelty of our results. Our main results are given in the two theorems below. In Theorem1we state a uniform Lipschitz estimate. In Theorem2we give a local error estimate and identify the building blocks of the regularity theory. Both results hold for weak solutions of the nonlinear equations (NS^ε) and hold without any smallness assumption on the size of the solutions.

Theorem 1(mesoscopic Lipschitz estimate). For allM ∈ (0,∞), there exists a constantε⁽¹⁾ ∈ (0,1) depending on kγkW^1,∞(R²) and M such that the following statement holds. For all ε ∈ (0, ε⁽¹⁾]andr∈[ε/ε⁽¹⁾,1], any weak solutionu^ε∈H¹(B^ε_1,+(0))³to(NS^ε)with

− ˆ

B^ε_1,+(0)

|u^ε|^{2 1}₂

≤M (3)

satisfies

− ˆ

B_r,+^ε (0)

|u^ε|^{2 1}₂

≤C_M⁽¹⁾r , (4)

where the constantC_M⁽¹⁾is independent ofεandr, and depends onkγk_W1,∞(R²)andM. Moreover, C_M⁽¹⁾is a monotone increasing function ofM and converges to zero asM goes to zero.

Remark1. (i) By using the Caccioppoli inequality in AppendixB, one can easily prove

− ˆ

B^ε_r,+(0)

|∇u^ε|^{2 1}₂

≤Cg_M⁽¹⁾

forr∈[ε/ε⁽¹⁾,¹₂]. Here the constantCg_M⁽¹⁾satisfies the same property asC_M⁽¹⁾.

(ii) In the paper [18], G´erard-Varet obtains a uniform H¨older estimate for weak solutions of the Stokes equations whenγ ∈ C^1,ω(R²)for a fixed modulus of continuityω. Let us emphasize that there is a gap in difficulty between the uniform H¨older estimate (right-hand side of (4) replaced by Cr^µwithµ∈(0,1)) and the uniform Lipschitz estimate (4). Indeed the Lipschitz estimate requires the analysis of the boundary layer corrector. Moreover, let us emphasize that the Lipschitz estimate is the best that can be proved foru^εuniformly inε. This comment does not contradict the uniform C^1,µestimate below. Indeed the estimate in Theorem2is a measure of the oscillation betweenu^ε and affine functions, and is not an estimate foru^εdirectly.

(iii) As in the works [8,18,31] one can combine the mesoscopic regularity estimate with the classical regularity, provided the boundary is regular enough, i.e. when∇γis H¨older continuous. In that case, we can prove the full Lipschitz estimatek∇u^εk_L∞(B^ε_1,+(0))for (NS^ε). However, one cannot expect such an estimate to hold in Lipschitz domains even for the Laplace equation with the Dirichlet boundary condition.

(iv) There is a version of Theorem1 for the linear Stokes equations; see Theorem3in Section4 below. An important application of such uniform Lipschitz estimates is for estimating the Green and Poisson kernels associated to the Stokes equations in the Lipschitz half-space{y3 > γ(y⁰)}.

Following [8,10], such estimates were proved for elliptic systems in bumpy domains in [31]. Such estimates play a crucial role for the homogenization of boundary layer correctors, in particular in the works [21,6,48].

Next let us state the result which gives a local justification of the Navier wall law. The following theorem is concerned with the polynomial approximation of weak solutions to (NS^ε) at mesoscopic

scales. Remark2below states consequences of the estimates in the theorem and Remark3estab- lishes the connection between our theorem and the Navier wall law.

Theorem 2(polynomial approximation). FixM ∈ (0,∞)and µ ∈ (0,1). Then there exists a constantε⁽²⁾ ∈(0,1)depending onkγk_W1,∞(R²),M, andµsuch that for all weak solutionsu^ε∈ H¹(B_1,+^ε (0))³to(NS^ε)satisfying the bound(3), the following statements hold.

(i)For allε∈(0, ε⁽²⁾]andr∈[ε/ε⁽²⁾,1], we have

− ˆ

B^ε_r,+(0)

u^ε(x)−

2

X

j=1

c^ε_r,jx3ej

2dx ¹₂

≤C_M⁽²⁾(r^1+µ+ε¹²r¹²), (5)

where the coefficientc^ε_r,j,j ∈ {1,2}, is a functional ofu^εdepending onε,r,kγk_W1,∞(R²),M, and µ, while the constantC_M⁽²⁾is independent ofεandr, and depends onkγk_W1,∞(R²),M, andµ.

(ii)We assume in addition thatγ ∈W^1,∞(R²)is2π-periodic in each variable. Then there exists a constant vector fieldα^(j)= (α^(j)₁ , α^(j)₂ ,0)^> ∈ R³,j ∈ {1,2}, depending only onkγk_W^1,∞_(R2)

such that for allε∈(0, ε⁽²⁾]andr∈[ε/ε⁽²⁾,1], we have

− ˆ

B_r,+^ε (0)

u^ε(x)−

2

X

j=1

c^ε_r,j(x3ej+εα^(j))

2dx ¹₂

≤Cg_M⁽²⁾(r^1+µ+ε³²r⁻¹²), (6)

where the coefficientc^ε_r,j, j ∈ {1,2}, is same as in the estimate(5), while the constant Cg_M⁽²⁾ is independent ofεandr, and depends onkγk_W1,∞(R²),M, andµ.

Remark2. (i) Each of the constantsC_M⁽²⁾andCg_M⁽²⁾satisfies the same property asC_M⁽¹⁾in Theorem1 as functions ofM.

(ii) Note that at the small scale, namely whenr=O(ε), the right-hand side in the estimate (5) is no better than the right-hand side of (4) in Theorem1. Hence there is no improvement at this scale. On the other hand, if we consider the caser∈[(ε/ε⁽²⁾)^δ,1]withδ∈(0,1), then we see that

r^1+µ+ε¹²r¹² ≤(1 + (ε⁽²⁾)¹²r^{1−δ2δ −µ})r^1+µ.

Therefore, we call the estimate (5) a mesoscopicC^1,µestimate at the scalesr∈[(ε/ε⁽²⁾)^δ,1]with δ∈(0,(2µ+ 1)⁻¹].

(iii) A comparison between the estimates (5) and (6) highlights the regularity improvement coming from the boundary periodicity. Indeed, the estimate (6) is sharper than (5) at mesoscopic scales becauseε³²r⁻¹² ≤ε¹²r¹² holds wheneverr∈[ε,1].

Remark3 (relation with the wall law). (i) Let us denote the polynomial in (6) byP_N,j^ε ,j∈ {1,2}:

P_N,j^ε (x) =x3ej+εα^(j). (7)

Then eachP_N,j^ε is a shear flow in the half-spaceR³+and is an explicit solution to the Navier-Stokes equations with a Navier-slip boundary condition

(NS^ε_N)











−∆u^ε_N +∇p^ε_N =−u^ε_N · ∇u^ε_N inR³+

∇ ·u^ε_N = 0 inR³+

u^ε_N,3= 0 on∂R³+

(u^ε_N,1, u^ε_N,2)^> =εM(∂₃u^ε_N,1, ∂₃u^ε_N,2)^> on∂R³+

with a trivial pressurep^ε_N = 0. Here the2×2 matrixM = (α^(j)_i )_1≤i,j≤2 can be proved to be positive definite; see Proposition11(ii). Thus the estimate (6) in Theorem2reads as follows: any weak solutionu^εto (NS^ε) can be approximated at any mesoscopic scale by a linear combination of the Navier polynomialsP_N,1^ε andP_N,2^ε multiplied by constants depending onu^ε. This is a local version of the Navier wall law at theO(ε^δ)-scales, which has been widely studied in the global framework.

(ii) Our result can be extended to the stationary ergodic or the almost periodic setting. We also note that the wall law breaks down when the boundary does not have any structure at all; see [20].

The novelty of our results can be summarized as follows:

(I) Singular boundary: it is just Lipschitz and has no structure (except in Theorem2(ii)).

(II) No smallness assumption on the size of solutions.

(III) Derivation of a local wall law and local error estimates.

As is stated in (I), one of the originalities of Theorem1is that it does not rely on the smoothness of the boundary such as, the H¨older continuity of∇γ. Moreover, one cannot use any Fourier methods due to the lack of structure of the boundary. In fact, when working with Lipschitz boundaries, the classical Schauder theory is not applicable directly since there is no improvement of flatness coming from zooming on the boundary as is explained in [32]. The smoothing happens at scales larger than that of the boundary layer thickness.

Concerning point (II), we are able to remove any smallness assumption on the size of the solutions in Theorem1 and Theorem2. This is in stark contrast with previous works concerned with the regularity of elliptic or Stokes systems [8,18,24,31,32]. Moreover, as far as we know the error estimates in the stationary global setting are all in the perturbative regime; see for instance [20].

Point (III) is concerned with Theorem2. It is important physically as well as mathematically since we are interested in the effects of rough boundaries on viscous fluids. Our result is a first-step toward understanding roughness effects on the Navier-Stokes flows in view of regularity improvement. As far as we know, estimate (6) is the first justification of a local wall law.

These three aspects are further discussed in connection with our strategy in the paragraph below.

Difficulties and strategy. The proof of Theorem1and Theorem2is based on a compactness ar- gument as in [31,32] originating from the works [8,9] on uniform estimates in homogenization.

In principle, we follow the strategy of [32] concerned with the regularity theory of elliptic systems in bumpy domains. The main points in [32] are: (1) construction of a boundary layer corrector in the Lipschitz half-space, (2) proof of the mesoscopic regularity by compactness and iteration. This strategy entails difficulties related to the lack of structure of the boundary which implies a lack of compactness of the solution to the boundary layer problem, and to the unavailability of Fourier me- thods up to the boundary. In addition to these difficulties, our proof is more involved due to: (i) the vectoriality of the equations (NS^ε) and the divergence-free condition, (ii) the nonlocal pressure, (iii) the nonlinearity of the Navier-Stokes equations and the lack of smallness of the solutions.

Concerning the first point, the (vectorial) divergence-free condition∇ ·u^ε= 0causes a difficulty in the compactness argument even for the Stokes equations; see Section4, especially Lemma13and its proof. A key idea is that no boundary layer is needed on the vertical component of the velocity.

Therefore the boundary layer corrector is naturally constructed as a divergence-free function.

Concerning the second point, let us stress a key difference between the stationary Navier-Stokes equations and the nonstationary ones. For the stationary Stokes equations imposed in a ballB₁(0), one can estimate the pressure directly in terms of the velocity as follows:

p−(p)_{B1(0) L2(B}

1(0))≤Ck∇pk_H−1(B₁(0))

≤Ck∆uk_H⁻¹_(B1(0))≤Ck∇uk_L2(B1(0)). (8)

Similar estimates in balls intersecting the boundary and for the Navier-Stokes equations are inten- sively used in our paper. This is in strong contrast with the nonstationary Navier-Stokes equations where the pressure interacts with the time derivative of the velocity. This yields parasitic solutions, which are responsible for a lack of local smoothing in time and also for a more serious lack of local smoothing in space of the gradient of the velocity in the half-space; see Kang [30] and Seregin and ˇSver´ak [46].

The third aspect is partly related to (ii). In typical statements of the partial regularity theory for the nonstationary Navier-Stokes equations, one assumes smallness of certain scale-critical quantities in εand hence one obtains linear equations in the limitε→0. Then the regularity theory for the linear equations yields a space-time H¨older regularity improvement for the original solution; see Lin [39], Ladyˇzenskaja and Seregin [37], and Mikhailov [42] for example. However, for the stationary Navier- Stokes equations discussed in our paper, we do not need such a smallness condition; see Theorem1.

The limit equations whenε→0are not linear, but we can prove the smoothness of weak solutions

becauseH¹bounds are enough to control both the nonlinear term and pressure term inL² space (see AppendixAfor details). Then bootstrapping using the standard elliptic regularity in a smooth domain leads to the (spatial)C^∞-regularity for the limit equations. Estimate (8) is the reason why one can bootstrap the regularity. Once the regularity is inherited at a fixed scaleθ∈(0,1), a serious difficulty arises in the iteration of such an estimate. At each step in the induction, we need to use the Caccioppoli inequality from AppendixBto control the normku^εk_L2. A naive approach yields an estimate that depends algebraically on the sizeM ofu^εas in (3). Hence the naive estimate becomes unbounded inM as the iteration proceeds. This prevents one from closing the induction due to the lack of uniformity. We overcome this difficulty by choosing the free parameterθin the compactness lemma in terms of the dataγandM. This is done in the spirit of the Newton shooting method. We will make this idea precise in Section5. It should finally be emphasized that the boundary layer corrector, entering the scheme for the nonlinear Navier-Stokes equations (NS^ε), solves the linear Stokes equations. This is expected from the following formal heuristics. Indeed, in the boundary layeru^ε'εv(x/ε), so thatvsolves−¹_ε∆v+εv· ∇v+∇q= 0,∇ ·v= 0.

Outline of the paper. The following two sections are devoted to the analysis of the boundary layer equations. In Section2we collect preliminary results on the well-posedness for the Stokes prob- lem and on the Dirichlet-to-Neumann operatorDNin the framework of non-localized Sobolev data.

In Section3we study the boundary layer equations by formulating equivalent equations on a strip bounded in the vertical direction and involving the nonlocal operator Dirichlet-to-NeumannDN. Our goal is to prove the unique existence of solutions of the equivalent equations. We study the asymp- totic behavior of the solution away from the boundary when the boundary is periodic in Subsection 3.3. In Section4we prove the linear version of Theorem1in order to show how the compactness method works in the regularity argument. In Section5 we prove the main results namely Theo- rem1and Theorem2. The regularity theory in a domain with a flat boundary and the Caccioppoli inequality are stated respectively in AppendicesAandB.

Notations. Let us summarize the notations in this paper for easy reference. Forx= (x1, x2, x3)^>∈ R³, we denote byx⁰its tangential part(x1, x2)^>. Ford∈ {2,3}andx, y∈R^d, we denote byx·y the inner product ofxandy. Then| · |denotes the corresponding norm inR^d. Forr ∈(0,1]and ε∈(0,1], we defineB^ε_r,+(0)andΓ^ε_r(0)as is done in (1) and set

Br(0) ={x∈R³|x⁰∈(−r, r)², x3∈(−r, r)}= (−r, r)³, Br,+(0) ={x∈R³|x⁰∈(−r, r)², x3∈(0, r)},

Γr(0) ={x∈R³|x⁰∈(−r, r)², x3= 0}.

Note that formally we haveBr,+(0) =B_r,+⁰ (0)andΓr(0) = Γ⁰_r(0). For an open setΩ⊂R³and a Lebesgue measurable functionf onΩ, we set

− ˆ

Ω

|f|= 1

|Ω|

ˆ

Ω

|f|, (f)Ω= 1

|Ω|

ˆ

Ω

f , (9)

where|Ω|denotes the Lebesgue measure ofΩ. Finally, we define the Sobolev-Kato spaceH_uloc^s (R²):

letϑ∈C₀^∞(R²)be such thatsuppϑ⊂[−1,1]²,ϑ= 1on[−¹₄,¹₄]², and X

k∈Z²

ϑk(x) = 1, x∈R², ϑk(x) =ϑ(x−k).

Then, fors∈[0,∞), we define the spaceH_uloc^s (R²)of functions of non-localizedH^senergy by H_uloc^s (R²) =n

u∈H_loc^s (R²) sup

k∈Z²

kϑ_kuk_Hs(R²)<∞o

and the spaceL²_uloc(R²)byL²_uloc(R²) =H_uloc⁰ (R²). We emphasize thatH_uloc^s (R²)is well-defined independently of the choice ofϑfor anys∈[0,∞)(see [3, Lemma 7.1] for the proof) and admits the embeddingW^1,∞(R²),→H_uloc^s (R²)whens∈[0,1).

Note that, since our interest is in the local boundary regularity of (NS^ε), the boundary condition is prescribed only on the lower part of∂B_r,+^ε (0). We work in the framework of weak solutions of

(NS^ε). A vector functionu^ε ∈H¹(B_1,+^ε (0))³is said to be a weak solution to (NS^ε) ifu^εsatisfies

∇ ·u^ε= 0in the sense of distributions,u^ε|Γ^ε₁(0)= 0in the trace sense, and ˆ

B_1,+^ε (0)

∇u^ε· ∇ϕ=− ˆ

B_1,+^ε (0)

(u^ε· ∇u^ε)·ϕ (10)

for anyϕ∈C_0,σ^∞(B_1,+^ε (0)). HereC_0,σ^∞(Ω)denotes the space of test functions{f ∈C₀^∞(Ω)³| ∇ · f = 0}whenΩis an open set inR³. For the pressurep^ε, we emphasize that the unique existence in L²(B^ε_1,+(0))up to an additive constant can be proved in a functional analytic way using the weak formulation (10); see a textbook [49, Lemma 3.3.1 and Remark 3.3.2, III] for details.

2. PRELIMINARIES

In this section we give preliminary results which will be used in the next section. In Subsection2.1 we prove a well-posedness result for the Stokes problem in the half-space with nonhomogeneous Dirichlet boundary data inH_uloc¹² (R²)³. Subsection2.2is devoted to the definition and basic prop- erties of the Dirichlet-to-Neumann operator onH

1 2

uloc(R²)³associated with the half-space problem.

Throughout this section, we use the Fourier transform and its inverse transform respectively defined by

F[f](ξ) = ˆf(ξ) = ˆ

R²

f(x)e^−ix·ξdx , ξ∈R², F⁻¹[f](x) = 1

(2π)² ˆ

R²

f(ξ)e^ix·ξdξ , x∈R²,

forf ∈ S(R²). We also use their extensions on the space of tempered distributionsS⁰(R²).

2.1. Analysis of the half-space problem. We consider the Stokes equations in the half-spaceR³+= {y= (y⁰, y₃)^>∈R³|y₃>0}with a non-localized boundary datau₀∈H_uloc¹² (R²)³

(SH)







−∆u+∇p= 0 inR³+

∇ ·u= 0 inR³+

u=u0 on∂R³+. The well-posedness of the problem (SH) is stated as follows.

Proposition 4.Letu0∈H

1 2

uloc(R²)³. Then there exists a unique weak solution(u, p)∈H_loc¹ (R³+)³× L²_loc(R³+)to(SH)satisfying

sup

η∈Z²

ˆ

η+(0,1)²

ˆ ∞ 0

|∇u(y⁰, y₃)|²dy₃dy⁰≤Cku0k²

H

12 uloc(R²)

(11) ,

whereCis a numerical constant.

Remark5. The pressurepcan be chosen to satisfy sup

η∈Z²

ˆ

η+(0,1)²

ˆ ∞ 0

|p(y⁰, y₃)|²dy₃dy⁰≤Cku0k²

H

1 2 uloc(R²)

.

Proof. We follow the proof of [20, Proposition 6] for the two-dimensional Stokes equations.

(Existence)We give only the outline here since this part is parallel to [20, Proposition 6]. Let u0∈H

1 2

uloc(R²)³. Then a solution to (SH) can be constructed by using the Poisson kernel(U, P)as u(y⁰, y3) =

ˆ

R²

U(y⁰−y˜⁰, y3)u0(˜y⁰) d˜y⁰, p(y⁰, y3) =

ˆ

R²

∇P(y⁰−y˜⁰, y3)·u0(˜y⁰) d˜y⁰, (12)

where the kernelsU =U(y)andP =P(y)are respectively defined by U(y) = 3y₃

2π(|y⁰|²+y₃²)⁵²





y₁² y₁y₂ y₁y₃ y₁y₂ y₂² y₂y₃ y1y3 y2y3 y₃²



, P(y) =− y3

π(|y⁰|²+y₃²)³² . (13)

We easily check thatuandpbelong toC^∞(R³+)by the derivative estimates of(U, P)

|∇^mU(y)| ≤ Cmy^δ₃^0m (|y⁰|²+y²₃)^{m+δ0m2 +2}

, |∇^m∇P(y)| ≤ Cm

(|y⁰|²+y₃²)^m+32

form ∈ N∪ {0}, which can be verified by direct computation. Hereδ0mdenotes the Kronecker delta. Moreover, we can prove the following estimates fora∈(0,∞)

sup

η∈Z²

ˆ

η+(0,1)²

ˆ ∞ a

|∇u(y⁰, y3)|²+|p(y⁰, y3)|² dy3dy⁰

≤ C a⁵ku₀k²_L2

uloc(R²), sup

η∈Z²

ˆ

η+(0,1)²

ˆ a 0

|∇u(y⁰, y3)|²+|p(y⁰, y3)|² dy3dy⁰

≤Cmax{1, a}ku₀k²

H

1 2 uloc(R²)

,

which lead to (11). We can also check that(u, p)solves (SH) andu=u₀on∂R³+in the trace sense.

(Uniqueness)Suppose thatu₀= 0in (SH). Then we aim at provingu= 0in the class sup

η∈Z²

ˆ

η+(0,1)²

ˆ ∞ 0

|∇u(y⁰, y₃)|²dy₃dy⁰ <∞. (14)

By the regularity theory of the Stokes equations and by the no-slip condition on∂R³+, we have sup

η∈Z²

ˆ

η+(0,1)²

ˆ ∞ 0

|∇^m∇u(y⁰, y3)|²+|∇^m∇p(y⁰, y3)|² dy3dy⁰

≤C sup

η∈Z²

ˆ

η+(0,1)²

ˆ ∞ 0

|∇u(y⁰, y3)|²dy3dy⁰ (15)

form ∈ N∪ {0}with a constantC depending onm; see the proof of [20, Proposition 6] for the two-dimensional case. Thus, for all fixedy₃∈(0,∞), we see thatu(·, y3)andp(·, y3)belong to the space of tempered distributionsS⁰(R²). Hence we can take the (partial) Fourier transform of (SH) withu₀= 0iny⁰. By lettingξ∈R²be the dual variable ofy⁰, we have the equations

(16)











(|ξ|²−∂₃²)ˆu⁰(ξ, y3) +iξp(ξ, yˆ 3) = 0, ξ∈R² (|ξ|²−∂₃²)ˆu3(ξ, y3) +∂3p(ξ, yˆ 3) = 0, ξ∈R² iξ·uˆ⁰(ξ, y3) +∂3uˆ3(ξ, y3) = 0, ξ∈R² ˆ

u(ξ,0) = 0. ξ∈R².

By eliminating the pressurep(ξ, yˆ 3)and using the divergence-free condition, we find (|ξ|²−∂₃²)²uˆ3(ξ, y3) = 0 in S⁰(R²).

(17)

To avoid the singularity atξ = 0, we introduce a functionϕ ∈ C₀^∞(R²)satisfyingϕ(ξ) = 0in a neighborhood ofξ = 0. Sinceϕ(ξ)ˆu3(ξ, y3)satisfies the equation (17) replacinguˆ3(ξ, y3)by ϕ(ξ)ˆu3(ξ, y3), there exist compactly supportedAi∈ S⁰(R²)andBi∈ S⁰(R²),i∈ {1,2}, such that

ϕ(ξ)ˆu3(ξ, y3) = (A1+y3A2)e^−|ξ|y3+ (B1+y3B2)e^|ξ|y3 in S⁰(R²).

The integrability in they₃ variable in (15) leads toB₁ = B₂ = 0, while the boundary conditions ˆ

u₃(ξ,0) =∂₃uˆ₃(ξ,0) = 0implyA₁ =A₂= 0. Hence we haveϕ(ξ)ˆu₃(ξ, y₃) = 0inS⁰(R²)for any cut-off functionϕ∈C₀^∞(R²)vanishing near the origin, which yields thatuˆ₃(ξ, y₃)is supported

atξ = 0. Thusu₃(y⁰, y₃)is a polynomial iny⁰ with coefficients depending ony₃, and therefore, because of (15) form= 0, we see thatu3(y⁰, y3)is in fact independent ofy⁰:

u₃(y⁰, y₃) =u₃(y₃). (18)

On the other hand, sinceϕ(ξ)ˆu₃(ξ, y₃) = 0inS⁰(R²), from the equations (16)₁and (16)3we have ϕ(ξ)|ξ|²p(ξ, yˆ ₃) = 0inS⁰(R²)for any cut-off functionϕ∈C₀^∞(R²)vanishing nearξ= 0. Thus, by a similar reasoning as foru₃, we conclude that there exists a functionf =f(y₃)such that

∇p(y⁰, y3) = (0,0, f(y3))^>. (19)

Then, going back to the original equations (SH) withu0 = 0, we see thatu⁰(y⁰, y3)solves the Laplace equation with the Dirichlet boundary condition

−∆u⁰= 0 inR³+

u⁰= 0 on∂R³+.

The Liouville theorem in the class (14) implies thatu⁰(y⁰, y3)is a constant vector field, and hence, u⁰ = 0by the boundary condition. Hence the proof will be complete if we prove u3 = 0. From

∆p= 0following from (SH), the equality (19) leads to∇p(y⁰, y3) = (0,0, a)^>with somea∈R. After inserting (18) and∂3p=ato (SH), we haveu3(y3) =ay₃²+by3+cwith some(b, c)∈R², which impliesu3=∂3p= 0from (15) and the boundary condition. This completes the proof.

2.2. The Dirichlet-to-Neumann operator. In this subsection we recall the definition and basic properties of the Dirichlet-to-Neumann operatorDNonH_uloc¹² (R²)³associated with the Stokes equa- tions (SH). We follow the procedure in [20, Subsection 2.2] treating the two-dimensional problem;

see also [3] studying the water-waves equations, [14] and [32] for related studies. Before going into the details, we give a useful lemma for estimating elements inH_uloc¹² (R²).

Lemma 6. Letu₀∈H_uloc¹² (R²)andχ∈C₀^∞(R²)withsuppχ⊂(−R, R)²for someR∈(0,∞).

Then we haveχu0∈H¹²(R²)and kχu0k

H¹²(R²)≤CRku0k

H

12 uloc(R²), where the constantCdepends only onkχkW^1,∞(R²).

We refer to the proof of [14, Lemma 2.26].

LetM = M(ξ)andM =e M(ξ)e be3×3matrices defined by M(ξ) =





|ξ|+ξ²₁|ξ|⁻¹ ξ1ξ2|ξ|⁻¹ iξ1

ξ1ξ2|ξ|⁻¹ |ξ|+ξ²₂|ξ|⁻¹ iξ2

−iξ1 −iξ2 2|ξ|



,

M(ξ) =e





0 0 iξ1

0 0 iξ2

−iξ1 −iξ2 0



. (20)

HereMis the symbol of the Dirichlet-to-Neumann operator of (SH) onH¹²(R²)³, whileMe is the singular part ofMbecause it is the Fourier transform of a derivative of a Dirac mass. Morever, let K=K(y⁰)be a3×3matrix defined by

K(y⁰) =F⁻¹[M−M](ye ⁰),

which must be understood in distributional sense: it is the inverse transform of a tempered distribu- tionM−M. From the theory of distributions, we see thate Kis a function onR²\ {0}satisfying

|K(y⁰)| ≤C|y⁰|⁻³. (21)

Then the operator DN on H

1 2

uloc(R²)³ of (SH) is defined in the following manner. Fix u0 ∈ H_uloc¹² (R²)³andR∈(1,∞), and letχ∈C₀^∞(R²)be a cut-off function such that

χ∈[0,1], suppχ⊂(−R−2, R+ 2)², χ= 1 in [−R−1, R+ 1]², kχk_W^1,∞_(R2)≤2.

Then we defineDN(u₀)as a functional on the set of test functionsϕsupported in(−R, R)² hDN(u0), ϕi_D⁰_,D=hF⁻¹[Mχud0], ϕi

H⁻¹2,H¹2

+ ˆ

R²

K∗(1−χ)u0

(y⁰)·ϕ(y⁰) dy⁰, (22)

whereh·,·iD⁰,D andh·,·i

H⁻¹2,H¹2 respectively denote the duality product between D⁰(R²)³ and D(R²)³ = C₀^∞(R²)³, andH⁻¹²(R²)³andH¹²(R²)³. Moreover,∗denotes the usual convolution product. Let us emphasize that the singular part of the Dirichlet-to-Neumann operatorMe does not appear in (22) becausesuppϕ∩supp(1−χ) =∅. Thanks to the properties ofχandϕ, the second term in the right-hand side of (22) converges, and consequently, the operatorDNgives an extension of the “standard” Dirichlet-to-Neumann operator onH¹²(R²)³. One can also check thatDNis well- defined independently of the choice ofχin a similar manner as in [20, Lemma 7]. We summarize the basic facts ofDNas follows.

Lemma 7. (i)Foru0∈H

1 2

uloc(R²)³andϕ∈C₀^∞(R²)³withsuppϕ⊂(−R, R)², we have

|hDN(u₀), ϕi_D⁰_,D| ≤CRku₀k

H

1 2 uloc(R²)

kϕkH¹2(R²), (23)

where the constantCis independent ofR∈(1,∞).

(ii)Letu0∈H_uloc¹² (R²)³and let{u0,n}^∞_n=1⊂H¹²(R²)³withsup_n∈Nku0,nk_L2(R²)<∞satisfy u_0,n * u₀ in H¹²((−k, k)²)

(24)

for allk∈N. Then forϕ∈C₀^∞(R²)³withsuppϕ⊂(−R, R)², we have

n→∞limhDN(u_0,n), ϕi_D⁰_,D=hDN(u₀), ϕi_D⁰_,D. (25)

(iii)Foru₀∈H_uloc¹² (R²)³andϕ∈C₀^∞(R³+)³with∇ ·ϕ= 0andsuppϕ⊂B_R(0), we have hDN(u0), ϕ|y3=0iD⁰,D =

ˆ

R³₊

∇u· ∇ϕ , (26)

whereu∈H_loc¹ (R³+)³is the weak solution to(SH)withu=u0on∂R³+provided by Proposition4.

In particular, ifu0is nonzero and compactly supported, then we have hDN(u0), u0i_D⁰_,D >0. (27)

Proof. (i) The first term in the right-hand side of (22) is estimated as

|hF⁻¹[Mχud0], ϕi

H⁻¹2,H¹2| ≤Ckχu0k

H¹2(R²)kϕk

H¹2(R²)

≤CRku0k

H

1 2

uloc(R²)kϕk

H¹2(R²), (28)

where Lemma6is applied in the second line. From the definition we have ˆ

R²

K∗(1−χ)u0

(y⁰)·ϕ(y⁰) dy⁰

= ˆ

R²

ˆ

R²

K(y⁰−y˜⁰)(1−χ(˜y⁰))u0(˜y⁰)·ϕ(y⁰) d˜y⁰dy⁰.

By using (21) and the properties ofχandϕ, we estimate the second term in the right-hand side of (22) as

ˆ

R²

K∗(1−χ)u0

(y⁰)·ϕ(y⁰) dy⁰

≤C ˆ

R²

ˆ

R²

1−χ(˜y⁰)

|y⁰−y˜⁰|³|u0(˜y⁰)||ϕ(y⁰)|d˜y⁰dy⁰

≤C ˆ

R²

ˆ

R²

1−χ(˜y⁰)

|y⁰−y˜⁰|³ d˜y^{0 1}₂ˆ

R²

1−χ(˜y⁰)

|y⁰−y˜⁰|³|u₀(˜y⁰)|²d˜y^{0 1}₂

|ϕ(y⁰)|dy⁰

≤Cku0k_L2 uloc(R²)

ˆ

R²

|ϕ(y⁰)|dy⁰

≤CRku0k_L²

uloc(R²)kϕkL²(R²), (29)

which with (28) implies the desired estimate (23).

(ii) From the assumption (24) we see that

n→∞limhF⁻¹[M\χu0,n], ϕi

H⁻¹2,H¹2 =hF⁻¹[Mχud0], ϕi

H⁻¹2,H¹2 . (30)

Fixk∈Narbitrarily. Then again from (24) we have

n→∞lim ˆ

R²

ˆ

|˜y⁰|≤k

K(y⁰−y˜⁰)(1−χ(˜y⁰)) u_0,n(˜y⁰)−u₀(˜y⁰)

·ϕ(y⁰) d˜y⁰dy⁰ = 0. (31)

On the other hand, if we choose k ∈ N to satisfyk > max{R+ 2,2R}, then χ(˜y⁰) = 0and 2|y⁰|<|y˜⁰|as long as|˜y⁰|> kand|y⁰| ≤R. Thus, in a similar way as in (29), we see that

ˆ

R²

ˆ

|˜y⁰|>k

K(y⁰−y˜⁰)(1−χ(˜y⁰)) u0,n(˜y⁰)−u0(˜y⁰)

·ϕ(y⁰) d˜y⁰dy⁰

≤C sup

n∈N

ku0,nk_L2(R²)+ku0k_L2 uloc(R²)

ˆ

R²

ˆ

|˜y⁰|>k

d˜y⁰

|˜y⁰|^{3 1}₂

|ϕ(y⁰)|dy⁰

≤ C k¹² sup

n∈N

ku0,nkL²(R²)+ku0k_L²

uloc(R²)

. (32)

Then from (31) and (32) we have

n→∞lim ˆ

R²

ˆ

R²

K(y⁰−y˜⁰)(1−χ(˜y⁰)) u_0,n(˜y⁰)−u₀(˜y⁰)

·ϕ(y⁰) d˜y⁰dy⁰ = 0, which with (30) implies the assertion (25).

(iii) Since the both sides of (26) are continuous with respect tou0∈H

1 2

uloc(R²)³, it suffices to prove it for smoothu0with all derivatives bounded. LetV andW denote the solutions to (SH) with the boundary dataχu0and(1−χ)u0. Then we haveu=V +W and one can check that

ˆ

R³₊

∇V · ∇ϕ=hF⁻¹[Mχud₀], ϕi

H⁻¹2,H¹2 , ˆ

R³+

∇W · ∇ϕ= ˆ

R²

K∗(1−χ)u0

(y⁰)·ϕ(y⁰) dy⁰.

Thus we obtain (26). Ifu0has a compact support, by choosingR ∈ (0,∞)sufficiently large, we see thatχu0=u0holds and(1−χ)u0is identically zero. Then from the definition ofDNwe have

hDN(u0), u0i_D⁰_,D =hF⁻¹[Muc0], u0i

H⁻¹2,H¹2.

By the definition ofMin (20), one can easily check that the right-hand side is positive as long asu₀ is nonzero. Hence we conclude (27). The proof of Lemma7is complete.